A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.
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2 views
The impact of jump on the returns of portfolio and asset pricing
There exsits jumps in financial market. What will be the impact of jump on the returns of portfolio and asset pricing?
Please explain it both academically and plainly. If you can give some excellent ...
1
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1answer
14 views
A theorem about the Poisson Point process.
In the proof of the Levy-Khintchine theorem, I saw a theorem about the Poisson
point process.
The theorem states that if $\Pi$ is a poission point process on $S$ with
intensity measure $\mu.$ Let ...
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0answers
16 views
Help me solve the invariant measure of $Q$
My $Q$ matrix is given by:
\begin{bmatrix}
-\lambda &0 &\lambda &0 &0 &... \\
\mu&-(\lambda+\mu) &0 &\lambda &0 &... \\
0&\mu ...
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1answer
40 views
Stochastic process, Gaussian, with zero mean is a Wiener process
Let $(\Omega, \mathcal F , \mathbb P)$ be a probability space and let $\mathcal F = \{\mathcal F_t\}_{t\ge} $ a filtration. Let $W=\{W_t;t ≥ 0\}$ be a stochastic process adapted to $\mathcal F$. ...
2
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0answers
29 views
Stopping times and $\sigma$-algebras
We have the usual $(\Omega, \mathcal{F}, P)$ stochastic basis. Let $\rho, \tau: \Omega \to T \cup \{+\infty\}$ be stopping times and $\mathcal{F}_{\rho}, \mathcal{F}_{\tau}$ their respective ...
2
votes
2answers
41 views
Chaos in finite field
Let's think about some finite field $\mathbb{F}$. Is it possible to construct a map
$x[n+1] = \mathcal{P}(x[n], x[n-1],...,x[n-k]), \ \ \ \forall x\in\mathbb{F} $
where $\mathcal{P}$ - ...
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votes
0answers
27 views
Prove $\mathbb{E}[X_t | \mathcal{F}_s] = \mathbb{E}[X_t | \sigma(\mathcal{F}_s \cup \mathcal{G}_s)] $
We want to prove that if $X_t$ is an $\mathcal{F}_t$ - martingale: $\mathbb{E}[X_t | \mathcal{F}_s] = X_s$ for $s<t$, then it's also a $\sigma(\mathcal{F}_s \cup \mathcal{G}_s)$- martingale. ...
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0answers
13 views
Levy process absolute moment
For a Levy process $(X_t)_{t\geq 0}$, we have $\mathbb{E}[X_t]=t\mathbb{E}[X_t^1]$ and $\text{Var}(X_t)=t\text{Var}(X_t^1)$. Does the same hold for the first absolute moment, i.e. does ...
1
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0answers
30 views
Construction binary tree
First let $\mu$ be the induced distribution of the random variable $X$ on $(\mathbb{R},\mathcal{B})$ and denote $EX=m$.
We also define for all $A\in G_{n+1}$ and $\omega\in X^{-1}(A)$
...
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votes
1answer
26 views
Examples of convergence of random variables
First, let's recall the definitions of 4 different types of convergence:almost surely, in $r$th mean, in probability and in distribution:
$X_n\xrightarrow{a.s.}X$ if $\{\omega \in ...
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0answers
10 views
Analytic tools in the theory of Galton-Watson processes
The questions basically aims at discussing the relative power of using probability generating functions, moment generating functions and characteristic functions as an example for ...
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1answer
36 views
Continuous Non negative martingale converging to 0
Is there any (non trivial) continuous non negative martingale which converges to 0?
1
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2answers
27 views
Generalization of Doob Dynkin for Stochastic processes
Let $\{X_t\}_{t\geq 0}$ be continuous time stochastic process and $\{\mathcal{F}_t^X\}_{t \geq 0}$ be the filtration generated by it. If the process $Y$ is $\{\mathcal{F}_t^X\}_{t \geq 0}$ adapted, is ...
3
votes
0answers
28 views
lower bound of expectation of stochastic differential equation
I'm looking for a lower bound on the expected value of a smooth, non-negative, increasing function $\mathbb{E}f(X_t)$, $f(0)=0$ of the solution to a stochastic differential equation $X_t = x + ...
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votes
0answers
17 views
Show $B_{t}^{2}$ is a weak solution of a stochastic differential equation. [closed]
Let $B_{T}$ be a Brownian motion in $\mathbb{R}$.
Show that $X_{t} = B_{t}^{2}$ is a weak solution of the stochastic differential equation
$dX_{t} = dt + 2\sqrt{|X_{t}|}d\tilde{B_{t}}$
where ...
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0answers
40 views
Exponential Levy process
We assume that the stochastic process L is a Levy process with the predictable characteristics triplet $(b,c,\nu)$. Which integrability conditions we should assume for the new stochastic process
...
3
votes
1answer
63 views
Optimal probability measure
Let $A$ be a finite set and let $\Bbb P$ be a probability measure on $A^{\Bbb N_0}$. Further, let $x_i:A^{\Bbb N_0}\to A$ be projection maps, so that $(x_i)_{i=0}^\infty$ can be treated as a ...
1
vote
1answer
39 views
How is Brownian motion predictable?
Could someone please explain how Brownian motion is predictable? My understanding is that a predictable process is one that depends on information up to time t say but not t itself, therefore W_t has ...
1
vote
2answers
51 views
Moment generating function of a stochastic integral
Let $(B_t)_{t\geq 0}$ be a Brownian motion and $f(t)$ a square integrable deterministic function. Then:
$$
\mathbb{E}\left[e^{\int_0^tf(s) \, dB_s}\right] = \mathbb{E}\left[e^{\frac{1}{2}\int_0^t ...
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votes
0answers
10 views
Distribution of partial sums of a $L^2$-transformed Gaussian Process
Our assumptions are: $X_t$ is a stationary sequence of standard normal random variables such that $\gamma _X (k)\sim L_{\gamma}(k)k^{2d-1}$ with $d \in (0,1/2)$, where $L_\gamma (k)$ is a slowly ...
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1answer
21 views
Proving weak existence of CIR process
Consider the following SDE
$$ X_t = x + \int_0^t \theta (\mu -X_s) ds + \int_0^t\kappa \sqrt{|X_s|} dW_s $$
where W is a brownian motion. I'm trying to show a weak solution exists, does anyone have ...
1
vote
1answer
45 views
finding the probability density function of $ dY_t = - Y_t X_t dW_t$
Could someone point me to where I can learn how to derive the stationary distribution for the martingale $Y_t$ which itself has stochastic volatility drive by $X_t$:
\begin{align}
dY_t &= - Y_t\ ...
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votes
1answer
20 views
Can an absorbing CTMC be reversible?
Can a CTMC with an absorbing state be reversible? I guess not, as the product of rates through any loop cannot be equal when the loop involves the absorbing state (Kolmogorov criterion). Is my ...
1
vote
1answer
34 views
The weighted distribution function for combination of two variables
For example, we have two random variables $a$ and $b$. And they have cumulative distribution function $F(x)$ and $H(x)$. We have number $0 < p < 1$.
Suppose, some machine get this random ...
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votes
1answer
35 views
Finding stationary Distribution
I need to know how to find the stationary distribution for this matrix:
$$
Q=
\begin{bmatrix}
-2 & 2 & 0 & 0 \\
1 & -2 & 1 & 0 \\
0 & 1 & -2 &1\\
0 & 0 ...
0
votes
1answer
66 views
Canonical Markov Process
Let $X$ be a canonical, right-continuous Markov process with values in a Polish state space $E$, equipped with Borel-$\sigma$-algebra $\mathcal{E}$ and we assume that $t\rightarrow E_{X_{t}}f(X_{s})$ ...
1
vote
0answers
22 views
Girsanov kernel moments
Let $Z_t=e^{\int_0^tq_tdB_t-\frac{1}{2}\int_0^tq^2_tdt}$, where $(q_t)_{t\geq0}$ is a predictable process, and $(B_t)_{t\geq0}$ a $\mathbb{P}$-Brownian motion. In particular, Novikov's condition ...
0
votes
0answers
20 views
Linear birth death process extinction probabilities
Given a birth and death process $X$ with $\lambda_n=n\lambda$ and $\mu_n=n\mu$ for $n\ge0$, and letting $P_n(t)=\Pr\{X(t)=n\}$, I need to prove that $P_0'(t)=\lambda P_0(t)^2-(\lambda+\mu)P_0(t)+\mu$.
...
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1answer
36 views
Continuous time Stochastic Process stopping time measurability
Let $\{X_t,\mathcal{F}_t;0\leq t < \infty\}$ be continuous time stochastic processes and $T$ be $\{\mathcal{F}_t\}_{0\leq t < \infty}$ stopping time. How to prove $X_T$ is $\mathcal{F}_T$ ...
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0answers
13 views
Solving the following SDE: dS=S(μdt+σe^(-t)dZ) from the BS-Model
I am trying to do an exercise where I have to solve the following stochastic Differential Equation, which is described by a modification of the Black-Scholes Model. It looks like the folllowing:
...
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votes
1answer
26 views
Brownian motion and convergence in probability of step functions
For positive $a$ and Brownian motion $B$, I want to compute $\int_0^a g(s)dB_s$ where $g \in L^2$ and $g$ is a step function if there exists partition $0=t_0 < ... < t_n = a$ such that $g = ...
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0answers
13 views
fGn asymptotic claim correlation
Let $(X_{i})$ be the fractional Gaussian noise for $H\in(0,1)$.
Since it is stationary $\mathbb{E}(X_{i}X_{j})$ only depends on $|j-i|$.
How can I prove for $\rho(|j-i|)=\mathbb{E}(X_{i}X_{j})$ that ...
1
vote
1answer
39 views
Moment generating function of two non-independent Brownian increments
I am writing to ask if it is possible to get closed-form solution to the expression to the following expression:
$\mathbb{E}[e^{\sigma^2(W_t-W_u)(W_s-W_u)}]$ where $W$ is a standard Brownian motion, ...
-1
votes
0answers
21 views
Random process x(t) =C and C is uniform over [-2,3]
I need reassurance that if I do a a few sample realizations of this random process they are all going to look the same. They are going to be an horizontal line with x(t) constant equal to 1/5.
I see ...
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0answers
55 views
Graduate research project in stochastic programming . [closed]
I don't know is this a good question or is this place is right to post this like question or not , but I need keen help, so I'm posting it.
I'm a graduate student & in this semester I've ...
0
votes
1answer
22 views
Integrating a Poisson Process with respect to itself
I am just learning about Poisson Processes and I feel somewhat comfortable with the basic concepts, but I am a little stuck with the following problem:
Let $N(t)$ be a Poisson process with intensity ...
1
vote
0answers
57 views
$dX_t=1_{X_t\not=0} dW_t$
Given The SDE : $dX_t=1_{X_t\not=0} dW_t$ with $ X_{0}=\xi $
how can I construct two obvious strong solutions to prove that SDE has non pathwise uniquenss
Indeed
Consider the stopping time $$ ...
3
votes
0answers
26 views
Orthogonal projections for minimization problem
I have trouble to understand the existence of a minimization problem in a Hilbert space. Let $(\Omega,\mathcal{F}_T,P)$ be a filtred probability space with filtration $(\mathcal{F}_t),0\le t\le T$. We ...
-2
votes
0answers
31 views
Transforming a Joint PDF [duplicate]
I have a pdf $f(X,Y)=(\frac{1}{4})^2e^{−\frac{(|x|+|y|)}{2}}$. My goal is to find the joint PDF $f(W,Z)$ taking in consideration this $W=XY$ and $Z=Y/X$.
I know I can not use Jacobian because is a ...
2
votes
0answers
40 views
Pure Birth Process
I encountered this problem while trying out various practice problems to study for my stochastic processes test. (It's not homework, it's just a practice problem).
Consider a pure birth process on ...
2
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0answers
51 views
Progressive measurability of stopped process
Let $(\mathcal{F}_t)_{t\in \mathbb{R}_+}$ be a filtration and let $X$ be a stochastic process progressively measurable with respect to $(\mathcal{F}_t)_{t\in \mathbb{R}_+}$. Let $T$ be a stopping time ...
2
votes
0answers
35 views
Product of predictable process and a characteristic function is integrable
Suppose the time parameter $t\in[0,T]$, $S$ is a Semimartingale and $\theta_t$ a predictable $S$-integrable process such that
$$\int_0^T\theta_u dS_u\ge -a$$
for a $a>0$. Furthermore ...
1
vote
2answers
68 views
Standard Brownian Motion
Let $\{X_t,t\ge 0\}$ be a standard Brownian motion. Compute the density of $X_t$ conditioned by $X_{t_1}$ and $X_{t_2}$ assuming that $t_1 <t<t_2$.
Can anyone give me some hint to start the ...
0
votes
0answers
14 views
Submartingale bounds
Let $X_1,X_2,\ldots$ be a submartingale with respect to the filtration generated by it. Is it possible to get any bounds for the probability $\mathbb{P}(X_2 < 0\mid X_1 >0)$ ?
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votes
1answer
51 views
Rewriting Markov process
Let $X$ be a Markov proces with state space $(E,\mathcal{E})$with initial distribution $\nu$ and transition function $P_{t}$, so $$E_{\nu}(f(X_{t+s})\mid\mathcal{F}_{s})=P_{t}f(X_{s})$$
Suppose that ...
-1
votes
1answer
101 views
Show that $M$ is a martingale
Let $B$ be typical Brownian motion with $\mu >0$ and $x \in \mathbb{R}$. $X(t):=x+B(t)+\mu t$, for each $t\geqslant 0$, Brownian motion with velocity $\mu$ that starts at $x$. For $r \in ...
1
vote
1answer
80 views
A Boundary crossing result for discrete brownian bridge
Let $S_n$ be a random walk with gaussian increments with $S_0=0$, i.e. $S_n-S_{n-1}\sim N(0,1), n\geq 1$. Fix $a>0,b\in \mathbb{R}$ and $c<a+bn$. Define the new process
$$
...
3
votes
0answers
58 views
Inadmissibility of Simpson's rule
Let $B_t$, $t\ge0$ be a standard Brownian motion and suppose $0<x_1<x_2<\cdots<x_n<1$. Then the conditional expectation
$$
\mathbb E\left(\int_0^1 B_t\,dt \,\middle\vert\, B_0, ...
0
votes
0answers
23 views
White Noise Process
Suppose $w_{t}$ is a normal white noise process. Is $z_{t} = w_{t}*w_{t-1}$ stationary?
Is my reasoning correct?
$Ew_{t}w_{t-1}w_{t+h}w_{t+h-1} = 0 $
for all $h$ implying that the series is ...
1
vote
0answers
21 views
Stochastic Processes, Doob's Inequality
please confirm if the following is correct.
Let $A = \{V_\tau > \epsilon\}$ and $\alpha = \min(\alpha_1,\tau)$, where $\alpha_1 = \min\{t\geq 0: X_t\geq \epsilon\}$ (So intuitively, $\alpha$ ...
